Multi-parameter families of K3 surfaces and hypergeometric functions
Multi-parameter families of K3 surfaces andhypergeometric functions
Fields InstituteWorkshop on Hodge Theory in String Theory
November 22, 2013
Andreas Malmendier, Colby College(work in progress with C. Doran)
Multi-parameter families of K3 surfaces and hypergeometric functions
1) Introduction
Examples of rigid rank-n local systems on P1\0, 1,∞Euler integral transform:
2F1
(1
2,
1
2; 1∣∣∣ t) .
=
ˆ 1
0
dx√x(x − t)
1√
1− x,
n+1Fn
(a1, a2, . . . , an,
12
c1, . . . cn−1, 1
∣∣∣ t) .=
ˆ 1
0
dx√x(x − t)
nFn−1
(a1, a2, . . . , anc1, . . . cn−1
∣∣∣ x)Families of twisted Legendre pencils:
E y21 = (1− x1) x1 (x1 − t) ,
K3 y22 = (1− x1) x1 (x1 − x2) x2(x2 − t) ,
CY 3 y23 = (1− x1) x1 (x1 − x2) x2(x2 − x3) x3 (x3 − t) .
Compute periods:ˆA
dx1
y1=
ˆ 1
0
dx1√x1(x1 − t)
1√
1− x1
.= 2F1
(1
2,
1
2; 1∣∣ t) ,
¨S
dx1 ∧ dx2
y2=
ˆ 1
0
dx2√x2(x2 − t)
ˆ 1
0
dx1
y1
.= 3F2
(12, 1
2, 1
21, 1
∣∣∣ t) ,
˚C
dx1 ∧ dx2 ∧ dx3
y3=
ˆ 1
0
dx3√x3(x3 − t)
¨S
dx1 ∧ dx2
y2
.= 4F3
(12, 1
2, 1
2, 1
21, 1, 1
∣∣∣ t) .
Multi-parameter families of K3 surfaces and hypergeometric functions
1) Introduction
Examples of rigid rank-n local systems on P1\0, 1,∞Euler integral transform:
2F1
(1
2,
1
2; 1∣∣∣ t) .
=
ˆ 1
0
dx√x(x − t)
1F0
(1
2;∣∣∣ t) ,
n+1Fn
(a1, a2, . . . , an,
12
c1, . . . cn−1, 1
∣∣∣ t) .=
ˆ 1
0
dx√x(x − t)
nFn−1
(a1, a2, . . . , anc1, . . . cn−1
∣∣∣ x)Families of twisted Legendre pencils:
E y21 = (1− x1) x1 (x1 − t) ,
K3 y22 = (1− x1) x1 (x1 − x2) x2(x2 − t) ,
CY 3 y23 = (1− x1) x1 (x1 − x2) x2(x2 − x3) x3 (x3 − t) .
Compute periods:ˆA
dx1
y1=
ˆ 1
0
dx1√x1(x1 − t)
1√
1− x1
.= 2F1
(1
2,
1
2; 1∣∣ t) ,
¨S
dx1 ∧ dx2
y2=
ˆ 1
0
dx2√x2(x2 − t)
ˆ 1
0
dx1
y1
.= 3F2
(12, 1
2, 1
21, 1
∣∣∣ t) ,
˚C
dx1 ∧ dx2 ∧ dx3
y3=
ˆ 1
0
dx3√x3(x3 − t)
¨S
dx1 ∧ dx2
y2
.= 4F3
(12, 1
2, 1
2, 1
21, 1, 1
∣∣∣ t) .
Multi-parameter families of K3 surfaces and hypergeometric functions
1) Introduction
Kummer pencil in Het/F-theory duality
Kummer Kum(E1 × E2) of two elliptic curves
Ei : y 2i = xi (1− xi )(xi − λi ), λi = λ(τi ), i = 1, 2.
(Isotrivial) elliptic fibration from projection to E1, useX = y 2
1 x2,Y = y 21 (y1y2) and t = x1 on base:
Y 2 = X(y1(t)2−X
) (X−λ2 y1(t)2
),
4 I∗0 , x1 = 0, 1, λ1,∞j(τF ) = j(τ2)
Multi-parameter families of K3 surfaces and hypergeometric functions
1) Introduction
Kummer pencil in Het/F-theory duality
Kummer Kum(E1 × E2) of two elliptic curves
Ei : y 2i = xi (1− xi )(xi − λi ), λi = λ(τi ), i = 1, 2.
(Isotrivial) elliptic fibration from projection to E1, useX = y 2
1 x2,Y = y 21 (y1y2) and t = x1 on base:
Y 2 = X (y 21 − X ) (X − λ2 y 2
1 ) ,
4 I∗0 , x1 = 0, 1, λ1
j(τF ) = j(τ2)
Periods of K3 surface (Picard rank 18):
Ωi =
‹Si
dt ∧ dX
Y=
˛A/B
dx1
y1
˛A/B
dx2
y2=
ω1ω2
τ1 ω1ω2
τ2 ω1ω2
τ1 τ2 ω1ω2
Picard-Fuchs equations: rank-4 linear system
2F1
(1
2,
1
2; 1∣∣λ1
) 2F1
(1
2,
1
2; 1∣∣λ2
)Period Domain:
[Ω1 : · · · : Ω4] ∈ P3
∣∣∣ Ω1Ω4 − Ω2Ω3 = 0Re(Ω1Ω4 − Ω2Ω3
)> 0
Multi-parameter families of K3 surfaces and hypergeometric functions
1) Introduction
Embedding of gauge theory into F-theory
Seiberg-Witten solution for N = 2 SU(2) gauge theory with fourquark flavors in d = 4.
Gauge coupling τG is encoded in rational elliptic fibration withsection over u-plane, called Seiberg-Witten curve.
Sen provided embedding of SW-curve into F-theory at limit point:
N = 2N = 2N = 2 gauge theory → F-theorytotal space: rational elliptic surface → elliptic K3 surfacefiberation: j(τG ) (isotrivial) = j(τF ) (isotrivial)sing. fibers: 2I ∗0 at u = 1,∞ 4 I ∗0 at t = 0, 1, λ1,∞VHS: elliptic curve period K3 periodslocal system: rank=1 → rank=2monodromy: (1,−1,−1) (T 2,T 2,−T 2)
periods: ω2√1−u = 1F0( 1
2 ; |u) · ω2
˛A
dt√t(t − λ1)
1F0(1
2; |u)︸ ︷︷ ︸
2F1( 12 ,
12 ;1|λ1)
·ω2
Multi-parameter families of K3 surfaces and hypergeometric functions
2) Rational elliptic surfaces
Rational surfaces
Rational elliptic surfaces S over CP1 with section:
S : y 2 = 4 x3 − g2 x − g3 ,g2 ∈ H0(O(4)),g3 ∈ H0(O(6)),
[t : 1] ∈ P1.
Consider extremal rational elliptic srfc with rk(MW) = 0,
classified by Miranda, Persson [’86].
Examples:
Legendre family over the t-line,y 2 = x(x − 1)(x − t)
Hesse pencil of cubics in P2,x3
1 + x32 + x3
3 − 3 t x1 x2 x3 = 0
Multi-parameter families of K3 surfaces and hypergeometric functions
2) Rational elliptic surfaces
Extremal rational surfaces and their periods
Rational elliptic surfaces S
S : y 2 = 4 x3 − g2 x − g3 ,g2 ∈ H0(O(4)),g3 ∈ H0(O(6)),
[t : 1] ∈ P1.
Extremal rational surfaces (up to ∗-transfer):
isotrivialI0 I ∗0 I ∗0I0 IV IV ∗
I0 III III ∗
I0 II II ∗
gen. modularI1 I1 I ∗4I2 I2 I ∗2I3 I1 IV ∗
I2 I1 III ∗
I1 I1 II ∗
modularI4 I2 I2 I4I2 I1 I1 I8I3 I3 I3 I3I9 I1 I1 I1I5 I1 I1 I5I6 I1 I2 I3
Multi-parameter families of K3 surfaces and hypergeometric functions
2) Rational elliptic surfaces
Extremal rational surfaces and their periods
Rational elliptic surfaces S
S : y 2 = 4 x3 − g2 x − g3 ,g2 ∈ H0(O(4)),g3 ∈ H0(O(6)),
[t : 1] ∈ P1.
Extremal rational surfaces (up to ∗-transfer):
isotrivialI0 I ∗0 I ∗0I0 IV IV ∗
I0 III III ∗
I0 II II ∗
gen. modularI1 I1 I ∗4I2 I2 I ∗2I3 I1 IV ∗
I2 I1 III ∗
I1 I1 II ∗
modularI4 I2 I2 I4I2 I1 I1 I8I3 I3 I3 I3I9 I1 I1 I1I5 I1 I1 I5I6 I1 I2 I3
Write down Picard-Fuchs first order linear system satisfied byperiods of dx
y and x dxy over cycles on the fibers:
~u =(ω =´At
dxy , η =
´At
x dxy
)
Multi-parameter families of K3 surfaces and hypergeometric functions
2) Rational elliptic surfaces
Extremal rational surfaces and their periods
Rational elliptic surfaces S
S : y 2 = 4 x3 − g2 x − g3 ,g2 ∈ H0(O(4)),g3 ∈ H0(O(6)),
[t : 1] ∈ P1.
Extremal rational surfaces (up to ∗-transfer):
gen. modular µI1 I1 I ∗4 1/2I2 I2 I ∗2 1/2I3 I1 IV ∗ 1/3I2 I1 III ∗ 1/4I1 I1 II ∗ 1/6
modular d qI4 I2 I2 I4 −1 0I2 I1 I1 I8 −1 0
I3 I3 I3 I31−i√
32
3−i√
32
I9 I1 I1 I11−i√
32
3−i√
32
I5 I1 I1 I58−5ϕ3+5ϕ
816+165ϕ(3+5ϕ)3
I6 I1 I2 I319
13
Solutions to Picard-Fuchs rank-2 linear system:ω = 2F1(µ, 1− µ; 1|t) ω = Hl(d , q; 1, 1, 1, 1|t)
Multi-parameter families of K3 surfaces and hypergeometric functions
3) Multi-parameter families of K3 surfaces
One-parameter families of K3 surfaces
Construction 1: quadratic twist with polynomial h
X1 = Sh : Y 21 = 4 X 3
1 − h2 g2 X1 − h3 g3
↓S : y 2 = 4 x3 − g2 x − g3 .
Quadratic twist adds 2 fibers of type I ∗0Parameter defines position of additional I ∗0 , h = t(t − A)
1-parameter families of lattice-polarized K3 surfaces (ρ = 19)
Example: TX = 〈2〉⊕2 ⊕ 〈−2〉, A 6= 0:
Esing I2 I2 I ∗2t 0 1 ∞︸ ︷︷ ︸
S is rational
Esing I ∗2 I2 I ∗2 I ∗0t 0 1 ∞ A︸ ︷︷ ︸
X1 is K3
Multi-parameter families of K3 surfaces and hypergeometric functions
3) Multi-parameter families of K3 surfaces
One-parameter families of K3 surfaces
Construction 1: quadratic twist with polynomial h
X1 = Sh : Y 21 = 4 X 3
1 − h2 g2 X1 − h3 g3
↓S : y 2 = 4 x3 − g2 x − g3 .
2 I ∗0 ’s, h = t(t − A), 2-form: dt ∧ dX1Y1
= 1√h(t)
dt ∧ dxy
Represent K3-periods as Euler transform of a HGF
Ω =‚
Sijdt ∧ dX1
Y1=´ t∗jt∗i
dt 1√h(t)
ω
They solve a 3rd oder ODE (=symmetric square of 2nd order).
Solutions to the rank-3 integrable linear system of K3 periods:
Ω = 3F2
µ, 12, 1−µ
1, 1
∣∣∣A Ω =[
Hl(
d , q4 ; 14 ,
34 , 1,
12
∣∣∣A) ]2
Multi-parameter families of K3 surfaces and hypergeometric functions
3) Multi-parameter families of K3 surfaces
One-parameter families of K3 surfaces
Construction 1: quadratic twist with polynomial h
X1 = Sh : Y 21 = 4 X 3
1 − h2 g2 X1 − h3 g3
↓S : y 2 = 4 x3 − g2 x − g3 .
2 I ∗0 ’s, h = t(t − A), 2-form: dt ∧ dX1Y1
= 1√h(t)
dt ∧ dxy
Represent K3-periods as Euler transform of a HGF
Ω =‚
Sijdt ∧ dX1
Y1=´ t∗jt∗i
dt 1√h(t)
ω
They solve a 3rd oder ODE (=symmetric square of 2nd order).
Solutions to the rank-3 integrable linear system of K3 periods:
Ω =[
2F1
(µ2, 1−µ
2; 1∣∣A)]2
Ω =[
Hl(
d , q4 ; 14 ,
34 , 1,
12
∣∣∣A) ]2
Multi-parameter families of K3 surfaces and hypergeometric functions
3) Multi-parameter families of K3 surfaces
One-parameter families of K3 surfaces
Proposition (M.-Doran)
There is a fundamental set of solutions [Ω1 : Ω2 : Ω3] such that
µ quadric surface series
1/2Ω2
1 + Ω22 − Ω2
3
2 Ω21 + 2 Ω2
2 − 2 Ω23
3F2
12 ,
12 ,
12
1, 1
∣∣∣A =∞∑n=0
(2n)!3
n!6An
26n
1/3 4 Ω21 + 3 Ω2
2 − 3 Ω23 3F2
13 ,
12 ,
23
1, 1
∣∣∣A =∞∑n=0
(2n)! (3n)!n!5
An
22n33n
1/4 4 Ω21 + 2 Ω2
2 − 2 Ω23 3F2
14 ,
24 ,
34
1, 1
∣∣∣A =∞∑n=0
(4n)!n!4
An
44n
1/6 Ω21 + 4 Ω2
2 − Ω23 3F2
16 ,
36 ,
56
1, 1
∣∣∣A =∞∑n=0
(6n)!n!3(3n)!
An
26n33n
First 4 cases with 4 singularities are obtained as double covers.
Cases 6 and 5 are related to Apery’s recurrence for ζ(3) and ζ(2):
Ω =∞∑n=0
(n∑
k=0
(nk
)2 (n+kk
)2)
An
4n , Ω =
( ∞∑n=0
n∑k=0
(nk
)2 (n+kk
)an)2
Multi-parameter families of K3 surfaces and hypergeometric functions
3) Multi-parameter families of K3 surfaces
One-parameter families of K3 surfaces
Construction 2: base change by double cover
X2 = S×C Bπ′
−−−−→ By yt=fA(s)
Sπ−−−−→ C
with t = (s+A/4)2
s we have ds ∧ dX2Y2
= 1√t(t−A)
dt ∧ dxy
Example (µ = 1/6): 1-param. family of K3 surfaces of Picardrank 19, TX = H ⊕ 〈−2〉, A 6= 0:
Esing I1 I1 II ∗
t 0 1 ∞︸ ︷︷ ︸S is rational
Esing I2 2 I1 2 II ∗
s −A/4 f −1A (1) 0,∞︸ ︷︷ ︸
X2 is K3
X2’s are one-parameter families with n = 1, 2, 3, 4, 5, 6, 8, 9and Mn = H ⊕ E8 ⊕ E8 ⊕ 〈−2n〉 lattice polarization.
Multi-parameter families of K3 surfaces and hypergeometric functions
3) Multi-parameter families of K3 surfaces
One-parameter families of K3 surfaces
Proposition (M.-Doran)
The two constructions give rise to degree-two rational mapsX2 99K X1 (ρ = 19) that leave the holomorphic two-form invariant.
The Picard-Fuchs equations of pairs X2,X1 coincide (rk=3).
Remarks:
The periods of the families with Mn lattice polarization forn = 1, 2, 3, 4, 6 agree with the results of Lian, Yau [’96],Dolgachev [’96], Verrill, Yui[’00], Doran [’00], andBeukers, Montanus, Peters, Stienstra [’84, ’85, ’86, ’00].One can “undo” the Kummer construction and provideinterpretation of K3 periods in terms of modular forms:
2F1
(µ2 ,
1−µ2 ; 1
∣∣A) = 2F1
(µ, 1− µ; 1
∣∣a), A = 4a(1− a),
Hl(
d , q4 ; 14 ,
34 , 1,
12
∣∣∣A) .= Hl
(d , q; 1, 1, 1, 1
∣∣∣a), A = quartic(a).
Multi-parameter families of K3 surfaces and hypergeometric functions
3) Multi-parameter families of K3 surfaces
Two-parameter families of K3 surfaces
Remarks:
Restrict (for simplicity) to case with 3-singular fibres.
Repeat above constructions with two parameters (A,B).
Example (µ = 1/6): M = H ⊕ E8 ⊕ E8-polarized case,
Esing 2 I1 2 I1 2 II ∗
s t(s) = 0 t(s) = 1 0, ∞︸ ︷︷ ︸X2 is M-polarized K3
99KEsing 2 I1 II ∗ 2 I ∗0t 0, 1 ∞ A,B︸ ︷︷ ︸
X1 is Kum(E1 × E2)
Other examples realize elliptic fibrations J3, J4, J6, J7, J11 onKum(E1 × E2) from Oguiso [’88].
Multi-parameter families of K3 surfaces and hypergeometric functions
3) Multi-parameter families of K3 surfaces
Two-parameter families of K3 surfaces
Set h(t) = (t −A)(t −B) in X1 and t = fA,B(s) in X2 (A 6= B) s.t.
dt ∧ dX1
Y1= ds ∧ dX2
Y2=
1√h(t)
dt ∧ dx
y
Proposition (M.-Doran)
The two constructions give rise to degree-two rational mapsX2 99K X1 (ρ = 18) that leave the holomorphic two-form invariant.
The Picard-Fuchs equations for pairs X2,X1 coincide.
K3-periods solve an integrable rank-4 linear system in ∂A, ∂B .
Fundamental solutions [Ω1 : Ω2 : Ω3, : Ω4] form a quadric in P3.
Multi-parameter families of K3 surfaces and hypergeometric functions
3) Multi-parameter families of K3 surfaces
Two-parameter families of K3 surfaces
Remarks:
HG local systemω(t) = 2F1(α, β; γ|t)
E.T.−→A-HG local system
Ω(A,B) =´ t∗jt∗i
dt√(t−A)(t−B)
2F1(t)
GKZ HG system (rk=2)c = (γ − β, α, β)l = [1,−1,−1, 1]t ω(t) = 0
−→
GKZ HG system (rk=4)C = (γ − β, 1/2, α, 1/2, 1 + α− β)
L =[
1 0 −1 −1 1 0 00 1 −1 0 0 −1 1
]~A,B Ω(A,B) = 0
Condition for non-resonance: α, β, γ − α, γ − β 6∈ ZCondition for quadrically related solutions: γ = 1, α + β = 1
∵ α = µ ∈ (0, 1), β = 1− µ, γ = 1
µtE .T .−→ ~A,B = µ/2
X µ/2Y
with XY = (A− B)2 and (1− X )(1− Y ) = (1− A− B)2.
Multi-parameter families of K3 surfaces and hypergeometric functions
3) Multi-parameter families of K3 surfaces
Two-parameter families of K3 surfaces
Remarks:HG local system
ω(t) = 2F1(α, β; γ|t)E.T.−→
A-HG local system
Ω(A,B) =´ t∗jt∗i
dt√(t−A)(t−B)
2F1(t)
GKZ HG system (rk=2)c = (γ − β, α, β)l = [1,−1,−1, 1]t ω(t) = 0
−→
GKZ HG system (rk=4)C = (γ − β, 1/2, α, 1/2, 1 + α− β)
L =[
1 0 −1 −1 1 0 00 1 −1 0 0 −1 1
]~A,B Ω(A,B) = 0
Condition for non-resonance: α, β, γ − α, γ − β 6∈ ZCondition for quadrically related solutions: γ = 1, α + β = 1
∵ α = µ ∈ (0, 1), β = 1− µ, γ = 1
µtE .T .−→ ~A,B = µ/2
X µ/2Y
Beukers [’13] determined full monodromy group for thissystem in terms of C.
Multi-parameter families of K3 surfaces and hypergeometric functions
3) Multi-parameter families of K3 surfaces
Three-parameter families of K3 surfaces
Use h(t) = (t − A)(t − B)(t − C ) to obtain 3-parameterfamilies X1 = Sh of K3 surfaces by quadratic twist.
Picard-Fuchs equations in A, B, C form a GKZ system:
resonant for µ = 12 , rk=5
non-resonant for µ 6= 12 , rk=6
There is one family where X1 is a 3-parameter family of K3surfaces with lattice polarization of Picard rank 17 : µ = 1
2 .
Families of curves from SW-theory:
I ∗0 + I ∗0extremal
mass=0←−−−− 3I2 + I ∗0rk(MW) = 1
RG−−−−→ 2 I2 + I ∗2extremal
K3 surfaces from twisting:Sen limit: 2-par. twist(I ∗0 + I ∗0 )Deformation: 2-par. twist(3I2 + I ∗0 ) = 3-par. twist(2I2 + I ∗2 )
Multi-parameter families of K3 surfaces and hypergeometric functions
3) Multi-parameter families of K3 surfaces
Kummer surfaces from SU(2)-Seiberg-Witten curve
Proposition (M.-Doran)
The family X1 = Sh → P1 (µ = 1/2) is a family of Jacobian K3surfaces of Picard rank 17.
There is a family X2 → P1 obtained from the covering mapt = (C s2 − B)/(s2 − 1).
X2 = Kum(A) where
ρ parameter A equation moduli space
17 A,B,C JacC (2) y2=x(x−1)(x−λ1)(x−λ2)(x−λ3) Γ2(2)\H2
18 A,B,C=∞ E1×E2 y2i =(2xi−1) [(4xi+1)2+9ri ] Γ\H×H
19 A,B=0,C=∞ E1×E1 y21 =(2x1−1)[(4x1+1)2+9r1] Γ0(2)\H
Mayr, Stieberger [’95], Kokorelis [’99]: moduli space ofgenus-two curves with level-two structure = moduli space ofN = 2 heterotic string theories compactified on K 3× T 2 with oneWilson line.
Multi-parameter families of K3 surfaces and hypergeometric functions
4) CY 3-folds and their periods
One-parameter families of Calabi-Yau 3folds
Construction 1: quadratic twist with polynomial h
X1 = Sh : Y 21 = 4 X 3
1 − h2 g2 X1 − h3 g3
↓S : y 2 = 4 x3 − g2 x − g3 .
h = u1 u22 (u1 − A u2) t1 (t1 − u1 t2) in [t1 : t2] and [u1 : u2]
Construction 2: birational family of 3-folds fibered byMn-polarized K3 surfaces (µ = 1
2 ,13 ,
14 ,
16 ↔ n = 4, 3, 2, 1)
Represent CY-periods as iterated Euler transform :
Ω =˝
C du ∧ dt ∧ dX1Y1
=´ u∗ju∗i
du√u(u−A)
´ t∗lt∗k
dt√t(t−u)
ω(t)
Solutions to the rank-4 integrable linear system of CY periods:
4F3
(µ, 1
2 ,12 , 1− µ
1, 1, 1
∣∣∣A
)µ = 1
2 ,13 ,
14 ,
16 give cases (m, a) = (16, 8), (12, 7), (8, 6), (4, 5)
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